Field of the Invention
This invention relates generally to the field of X-ray diffraction crystallography and, more specifically, to the measurement of lattice parameters in an X-ray diffraction crystallography experiment.
Description of the Related Art
When a beam of radiation with wavelength on the order of the spacing between atoms is made incident upon a crystalline material, several interferometrically reinforced beams are emitted from the sample when the proper geometry of the incident beam relative to the spacing of interest is attained. The condition in which diffraction occurs was described by Bragg as λ=2d sin θ, where λ represents the wavelength of radiation used, d represents the interatomic spacing and θ represents the angle at which the beam is made incident upon the crystal. To reach the diffracting condition for a specific crystallographic plane of interest, knowledge of the crystal system must be combined with knowledge of the motion of a goniometer in which the crystal is mounted, a method described by Paul Ewald with his construction of the Ewald Sphere in Reciprocal Space.
In an Ewald Sphere construction, the diffracting condition is represented by a sphere of radius 1/λ. This sphere intersects the origin of reciprocal space at one point on its surface. The reciprocal lattice, in which the Ewald Sphere is constructed, has axes which are related to the real space distance between atomic planes through an inverse relationship. The real space motions of an X-Ray Diffractometer result in the Ewald sphere being rotated in an analogous fashion in reciprocal space. When a reciprocal lattice point, whose shape and location are defined by the structure of the crystalline sample, in reciprocal space intersects the Ewald sphere, the condition is met such that a reinforcement of the scattered radiation (with wavelength equal to that of the incident beam) leaving the sample occurs. This is commonly referred to as a “reflection.” By manipulating the orientation of a detector relative to the crystal, that beam can be captured, and its relative coordinates used to determine the atomic spacings in the material. It is common practice to collect an extensive number of these data points, and map them in what is called a “reciprocal space map.”
In conventional systems, the construction of a reciprocal space map makes use of either a point detector or a one-dimensional (1D) detector to collect the data related to the reciprocal lattice. Thus, for each orientation of the sample, the detector is moved relative to the sample to cover all regions of interest where there might be a reflection. Once reflection data has been gathered over a large range of orientations, a reciprocal space map may be assembled. However, depending on the number of points being sampled, the process may take hours, or even days, to complete.
While crystal samples may take different forms, one particular structure of interest is a material having two different crystal layers, such as a crystal substrate with a film of crystal material deposited on it. For a structure such as this, rotation of the sample (and corresponding rotation of the Ewald sphere) results in reflections being generated from both the substrate layer and the film layer. By measuring several reflections associated with the crystal structure of the substrate and film, properties of the real space crystal structure, such as the spacing of atoms normal to the surface of the crystal, spacing of the atoms in the plane of the surface of the crystal and the relationship of the film crystal structure to the substrate crystal structure, can be determined. This is done conventionally with a zero-dimensional (point) or one-dimensional (line) detector, which collects a series of points that are post-processed into a planar cross-sectional map through reciprocal space for each of the materials.